Mellin transform with compact support

Question

Mellin transform for $f(x)$ is usually defined as:

$$F(s)=\int_0^\infty f(x)x^{s-1}dx$$

Is there a Mellin transform with compact support?

For example like $$F(s,a,b)=\int_a^{b} f(x)x^{s-1}dx,\qquad 0\le a \lt b \le \infty$$